Chapter 2.4, Problem 53E

To determine

To find: The slope of the curve $y=\sin x$ at $x=1$.

Answer to Problem 53E

The slope of the curve $y=\sin x$ at $x=1$ is equal to $0.54$.

Explanation of Solution

Given information: The curve is $y=\sin x$.

Calculation:

Check the slope of the function at.

The slope of the function at $x=1$ can be calculated as:

  $\underset{h\to 0}{\lim }\frac{f\left(1+h\right)-f\left(1\right)}{1+h-1}=\underset{h\to 0}{\lim }\frac{f\left(1+h\right)-f\left(1\right)}{h}$

Substitute $1+h$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\sin x\\ f\left(1+h\right)=\sin \left(1+h\right)\end{array}$

Substitute $1$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\sin x\\ f\left(1\right)=\sin \left(1\right)\end{array}$

Substitute $\sin \left(1+h\right)$ for $f\left(h\right)$ and $\sin 1$ for $f\left(0\right)$ in the formula of slope.

  $\underset{h\to 0}{\lim }\frac{f\left(1+h\right)-f\left(1\right)}{h}=\underset{h\to 0}{\lim }\frac{\sin \left(1+h\right)-\sin 1}{h}$

Check the limit of difference quotient as $h\to 0$ graphically.

To graph a function $y=\frac{\sin \left(1+h\right)-\sin 1}{h}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{\sin \left(1+h\right)-\sin 1}{h}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{SIN}}\boxed{1}\boxed{+}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{\text{SIN}}\boxed{1}\boxed{)}\boxed{)}\boxed{÷}\boxed{\text{X}}$.

The display will show the equation,

  ${\text{Y}}_{\text{1}}=e^\left(1\right)\left(\left(e^\left(\text{X}\right)-1\right)/\text{X}\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from the graph, the function tends to $0.54$ as $h$ tends to $0$.

Therefore, the slope of the curve $y=\sin x$ at $x=1$ is equal to $0.54$.